Multiplying two matrices together is most easily explained with an example. Once we work through the example, we’ll express what we did in words.

Imagine we have two matrices, A and B. A has 5 rows and 3 columns. B has 3 rows and 4 columns. Multiplying A * B results in C, which has 5 rows and 4 columns.

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The numbers down the left of each matrix are the row numbers. The ones across the top are the column numbers.

We refer to an element in a matrix by its row and column position:

Matrix Name[row, column]. A[1,1] is 1. B[2,3] is 6. C[5,4] is… well, we don’t know what C[5,4] is.

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The shaded cells are the elements referred to.

To calculate C[1,1], we need the first row of A and the first column of B.

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We need to take the dot product of the first row of A and the first column of B. The dot product is represented by a dot: x • y. Dot product means multiplying each element of A’s first row by the corresponding element of B’s first column, then adding up the result of the multiplications. Put another way:

A's Row 1 • B's Column 1 = A[1,1] * B[1,1] + A[1,2] * B[2,1] + A[1,3] * B[3,1]

It’s easier to calculate if we write it down. First, we’ll take just the row and column we need.

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The numbers down the left are the row numbers.
The ones across the top are the column numbers
.

Next, we’ll write B’s first column underneath A’s first row, lining up A’s columns with B’s rows.

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The numbers across the top represent column numbers for A and row numbers for B.

Then we’ll multiply the numbers in each column and write each result underneath. 1 * 12 is 12, 2 * 8 is 16, and 3 * 4 is 12.

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Finally, we add them together. 12 + 16 + 12 = 40

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And put the result in C[1,1].

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The remaining elements of C are calculated in a similar manner. C[1,2] is the dot product of the first row of A and the second column of B. C[1, 3] is the dot product of the first row of A and the third column of B.

Even though A only has three columns, we’re taking the dot product of rows of A and columns of B. Since B has another column, we take the dot product of the first row of A and the fourth column of B to calculate C[1, 4]. If B had additional columns, we would continue calculating dot products using the first row of A until we ran out of columns.

The remaining rows of A follow the same pattern. C[2,1] is the dot product of the second row of A and the first column of B. C[3,2] is the dot product of the third row of A and the second row of B. And so forth.

The general rule is C[i, j] equals the dot product of the i-th row of A and the j-th column of B. We’ll do one more example. C[5, 3] is the dot product of the fifth row of A and the third column of B.

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The dot product of A’s row 5 and B’s row 3 is 244.

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So C[5, 3] is 244. Let’s fill in the rest of C.

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Matrix Multiplication Formula

Taking what we learned from the above example:

If A is a matrix with M rows and P columns and B is a matrix with P rows and N columns, A * B results in a matrix C. C has M rows and N columns. Each element in C, C[i, j], is the dot product of the i-th row of A and the j-th column of B. Two matrices can be multiplied if and only if A has the same number of columns that B has rows.